Unit 5Section 5-3

5-3: Mean, Variance, Standard Deviation, and Expectation Determining the mean, standard

deviation, and variance of a probability distribution is calculated differently from samples.

In this section we will also look at another measure known as expectation.

Steps for the Mean of a Probability Distribution

Determine your probability distribution

Multiply each possible outcome by its corresponding probability

Add the products up

The result is the mean of the probability distribution

Section 5-3

Example 1: Find the mean of the number of

spots that appear when a die is tossed.

Section 5-3

Example 2: In a family with two children, find

the mean of the number of children who will be girls.

Section 5-3

Example 3: If three coins are tossed, find the

mean of the number of heads that occur.

Section 5-3

Steps for the Variance of a Probability Distribution

Determine your probability distribution

Determine your mean for the probability distribution

Multiply the square of each outcome by its corresponding value, then sum up the products.

Subtract the square of the mean from the value you calculated.

The result is the variance of the probability distribution

Section 5-3

Steps for the Standard Deviation of a Probability Distribution

Determine your probability distribution

Determine your mean for the probability distribution

Determine your variance for the probability distribution

Take the square root of the variance.

The result is the standard deviation of the probability distribution

Section 5-3

Section 5-3

Example 4: Find the variance and standard

deviation for Example 1.

Section 5-3

Example 5: Five balls numbered 0, 2, 4, 6, and

8 are placed in a bag. After the balls are mixed, one is selected, its number noted, and replaced. If this experiment is repeated many times, find the variance and standard deviation of the numbers on the balls.

Section 5-3

The expected value of a discrete variable of probability distribution is the theoretical average of the variable.

The symbol E(X) is used for expected value

Expected value is often used in games of chance, insurance, etc.

Expected value is calculated in the same way you would calculate theoretical mean.

Section 5-3

Example 6: One thousand tickets are sold at

$1 each for a color television valued at $350. What is the expected value of the gain if a person purchases one ticket?

Section 5-3

Example 7: A financial advisor suggests that

his client select one of two types of bonds in which to invest $5000. Bond X pays a return of 4% and has a default rate of 2%. Bond Y has a 2.5% return and default rate of 1%. Find the expected rate of a return and decide which bond would be a better investment. When the bond defaults, the investor loses all the investment.

Homework: Pg 253: 1 - 7

Section 4-5